What is the general form of a second order differential equation?

Hi, I know this is more of a maths question than an EE question, but I always get brilliant answers from this site so I thought I'd ask here.
It's only a quick question:

If the general form of a second order linear non homogeneous differential equation is

x'' + 2ζωx' + ω²x = Ku(t)

what do ζ, ω and K represent?

I'v been told ζ is the damping ratio, and ω is the un-damped natural frequency, but what does K represent?

I study electrical engineering and we have been told that K is the steady state gain, but this cant be true. If you do a Laplace transform on this equation and set s to zero you see that the steady state gain is K/ω².

Also if you solve the differential equation it is clear that the steady state gain is K/ω².

So why did my tutor tell me that K is the steady state gain?

I have also been told that to find the frequency at which the system will oscillate at, the formula is
ω1=ω(1-ζ^2)^0.5

Is this true?

Thanks!

Well you don't normally see this in the form you have, but that isn't a big deal. Normally you see the X" with the the radian frequency as the dividend.

Example:

0 = 1/( S^2(/ω^2) + S/(ω*Q) + 1)

This is the natural solution, and part of the total solution. But this is just algebra. If you get where you are, then you are pretty much home free.

Anything else I can help with? This is good stuff!
===========================================================================================
I’v been told ζ is the damping ratio, and ω is the un-damped natural frequency, but what does K represent?

ζ , yes referred to as the damping ratio. Actually you set the characteristic equation to zero 1st. There are three solutions determined by the discriminate, using the quadratic equation, real roots, imaginary roots and complex conjugates.

So K in this case is a constant and defines the amplitude at t0 (to = time zero).